A Szeg\"o limit theorem for a class of Toeplitz operators on the Bergman space of the unit ball with singular symbols

Abstract

We obtain a Szeg\"o limit theorem for a family of Toeplitz operators defined on the weighted Bergman space of the unit ball $\mathbb{B}_{n}$. The symbols of these operators are supported on some isotropic or co-isotropic submanifold $\Gamma \subseteq \mathbb{B}_{n}$ and can be seen, in general, as measures that are singular with respect to the Lebesgue measure on $\mathbb{C}^{n}$. The given theorem allows to describe the asymptotic behavior of these operators as the parameter of the weighted Bergman space tends to infinity.